Optimal. Leaf size=178 \[ \frac{2 a (24 A+28 B+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a (24 A+28 B+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{7 d} \]
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Rubi [A] time = 0.550503, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4265, 4086, 4015, 3805, 3804} \[ \frac{2 a (24 A+28 B+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a (24 A+28 B+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{7 d} \]
Antiderivative was successfully verified.
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Rule 4265
Rule 4086
Rule 4015
Rule 3805
Rule 3804
Rubi steps
\begin{align*} \int \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 A \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{7 d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)} \left (\frac{1}{2} a (A+7 B)+\frac{1}{2} a (4 A+7 C) \sec (c+d x)\right )}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{7 a}\\ &=\frac{2 a (A+7 B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d \sqrt{a+a \sec (c+d x)}}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{7 d}+\frac{1}{35} \left ((24 A+28 B+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a (24 A+28 B+35 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (A+7 B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d \sqrt{a+a \sec (c+d x)}}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{7 d}+\frac{1}{105} \left (2 (24 A+28 B+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{4 a (24 A+28 B+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{2 a (24 A+28 B+35 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (A+7 B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d \sqrt{a+a \sec (c+d x)}}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.379889, size = 105, normalized size = 0.59 \[ \frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} ((141 A+28 (4 B+5 C)) \cos (c+d x)+6 (6 A+7 B) \cos (2 (c+d x))+15 A \cos (3 (c+d x))+228 A+266 B+280 C)}{210 d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.373, size = 120, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( dx+c \right ) \right ) \left ( 15\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+18\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+21\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+24\,A\cos \left ( dx+c \right ) +28\,B\cos \left ( dx+c \right ) +35\,C\cos \left ( dx+c \right ) +48\,A+56\,B+70\,C \right ) }{105\,d\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}}\sqrt{\cos \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.31795, size = 686, normalized size = 3.85 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.488651, size = 284, normalized size = 1.6 \begin{align*} \frac{2 \,{\left (15 \, A \cos \left (d x + c\right )^{3} + 3 \,{\left (6 \, A + 7 \, B\right )} \cos \left (d x + c\right )^{2} +{\left (24 \, A + 28 \, B + 35 \, C\right )} \cos \left (d x + c\right ) + 48 \, A + 56 \, B + 70 \, C\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{105 \,{\left (d \cos \left (d x + c\right ) + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt{a \sec \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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